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how to calculate remainder of $7^ {154}$ when it is divided by $341$. Could you please state which method or theorem to use.

Andronicus
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Abinash Patra
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  • Hint $ $ By Fermat $,7^{30}\equiv 1\bmod 11\ &\ 31,,$ so also $!\bmod 341,,$ so $\bmod 341!:\ 7^{154}\equiv 7(7^3)\equiv 7(2),$ by $,154\equiv 4\pmod{!30}$ and modular order reduction. – Bill Dubuque Mar 21 '20 at 16:09
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    Fermat's Little Theorem, and the Chinese Remainder theorem. To get started $341 = 31\times 11$ and $31$ and $11$ are both prime. – fleablood Mar 21 '20 at 16:10
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    You can also use Eulers theorem and educated guessing with successive squaring. $\phi(341)=\phi(31)\phi(11) = 300$ so $7^{150}$ is probably $\equiv \pm 1\pmod {341}$ and testing $7^k$ were $k|150$ particularly $k=3,5,10$ will likely be useful. $7^2=49$ and $7^3=(50-1)7=350-7=343\equiv 2$. $7^{30}\equiv 1024\equiv 1$. So $7^{150}\equiv 1$. – fleablood Mar 21 '20 at 16:20

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Notice, that $7^3 = 343$ and $341*3=1023$, so $2^{10} \equiv 1 \pmod{341}$, then

$$7^{154} \equiv 7 * 2^{51} \equiv 7 * 1 * 2 = 14 \pmod{341}$$

Andronicus
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